MATH 280 Modern AlgebraFall 2015 · Syllabus

Class InformationInstructor: Dr. Lauren WilliamsClass Meeting: MWF 9:15 - 10:20 in Hirt 209Office: Old Main 401 (Tower)Office Phone: (814) 824-2226Office Hours: Mon 10:30-12, Tues 12-1, Wed 2:15-3:15, Thurs 11-12, Fri 10:30-12Email: lwilliams2@mercyhurst.eduWebsite: http://math.mercyhurst.edu/~lwilliams

Course DescriptionThis is the first semester of a year long sequence on the study of algebraic structures. Course topics include the prop-erties of numbers, equivalence relations, groups, rings, fields, direct products, homomorphisms and isomorphisms,and the natural development of various number systems.

Course ObjectivesOn successful completion of the course, students should be able to:

• provide the definitions of algebraic objects, and know some examples of each.

• develop abstract and critical reasoning by studying and writing mathematical proofs.

• understand the connection between modern algebra and other branches of mathematics.

• relate the material learned in this course to prerequisite courses.

• recognize algebraic structures and objects in everyday situations.

• learn about the historical development of modern algebra.

TextbookContemporary Abstract Algebra, by Joseph Gallian, 8th Edition (older editions are fine too). No other supplies arerequired for the course.

Depiction of a non-abelian group of order 43,252,003,274,489,856,000

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Lectures and NotesPartial notes for each lecture, including definitions and theorem statements, will be available on the course website.You should read the relevant course notes for each topic before class, and you’re encouraged to come to class withquestions. By seeing the definitions and theorems ahead of time, you’ll be able to focus on understanding them inclass, rather than concentrating on writing.

The notes you’ll be given are not complete, but will complement the examples and proofs of important theoremswe’ll cover in class. How you choose to use these notes is up to you. One suggestion for studying is to recopy allnotes (handouts and those you take in class) in a way that’s clear to you. As you’re doing so, you’re likely to uncoverquestions you hadn’t thought of before.

If you miss a class, make sure you get the additional notes from a classmate. Attendance is not required nor part ofyour grade, but coming to class regularly is in your best interest.

HomeworkYou will have several assignment due throughout the semester (generally every Friday). You should expect to spenda fair amount of time on each assignment - don’t wait until the night before it’s due to get started! You are freeto work together on your assignments, but everyone must submit their own work, in their own words. The lowesthomework grade will be dropped. Late homework will not be accepted, unless you talk to me before it’s due.

ExamsWe will have a midterm exam and a final exam. The midterm exam will have a take home portion, followed by an inclass portion. Unlike the homework assignments, you are not permitted to work together on the take home portionof the midterm exam. The in class portion will be comprised of shorter questions based on definitions, while the takehome will include some proofs. The final exam will be a cumulative exam with no take home portion.

Exam Dates:Midterm Exam (In Class and Take Home Portion Due): Monday, October 19Final Exam: Friday, December 11, 8:00-10:00 am

Final GradesGrades will be calculated as follows:

60% - Average of homework assignments (lowest grade dropped)20% - Midterm Exam20% - Final Exam

Grading scale:

F D D+ C C+ B B+ A0-59 60-64 65-69 70-77 78-83 84-89 90-93 94-100

Mapping of the exceptional simple Lie group E8

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Additional Resources on the Course WebsiteThe site for this course contains lots of useful material and information, including:

• all information found in this syllabus

• a detailed (and updated) course schedule

• lecture notes for each class meeting, posted at least one week before the meeting

• copies of any course handouts

• applets designed for our course that help reinforce definitions and concepts

• a customized study guide, with all definitions and theorems, outlines of proofs, and many practice questionswith solutions

• links to additional resources on the web, such as free algebra textbooks and study materials

• suggestions and tutorials for software related to modern algebra

• a calculator to determine what grade you’ll need on the final exam to earn your desired final grade, based onyour homework and midterm grades

Support of the Mercy MissionThis course supports the mission of Mercyhurst University by creating students who are intellectually creative.Students will foster this creativity by: applying critical thinking and qualitative reasoning techniques to new disciplines;developing, analyzing, and synthesizing scientific ideas; and engaging in innovative problem solving strategies.

Learning DifferencesIn keeping with college policy, any student with a disability who needs academic accommodations must call LearningDifferences Program secretary at 824-3017, to arrange a confidential appointment with the director of the LearningDifferences Program during the first week of classes.

Cayley table for dihedral group of order 12

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Math 280 Modern Algebra Course Schedule - Fall 2015

Aug 26 Topic: Course Intro, Sets

Aug 28 Topic: Properties of Numbers, Modular Arithmetic

Aug 31 Topic: Logic, Proofs

Sept 2 Topic: Linear Algebra

Sept 4 Topic: Functions, Equivalence Relations Assignment 1 DueSept 7 Labor Day - NO CLASS

Sept 9 Topic: Binary Operations, Groups

Sept 11 Topic: Groups, Properties of Groups Assignment 2 DueSept 14 Topic: Properties of Groups, Order

Sept 16 Topic: Subgroups

Sept 18 Topic: The Dihedral Group Assignment 3 DueSept 21 Topic: Cayley Tables

Sept 23 Topic: Centers and Centralizers

Sept 25 Topic: Cyclic Groups Assignment 4 DueSept 28 Topic: Cyclic Groups

Sept 30 Topic: Permutations

Oct 2 Topic: The Symmetric and Alternating Groups Assignment 5 DueOct 5 Topic: The 15 Puzzle and The Rubix Cube Group

Oct 7 Topic: Group Homomorphisms

Oct 9 Topic: Group Homomorphisms Assignment 6 DueOct 12 Topic: Group Isomorphisms

Oct 14 Topic: Group Isomorphisms Assignment 7 DueOct 16 Mid-Semester Break - NO CLASS

Oct 19 Midterm Exam Take Home Portion DueOct 21 Topic: Cosets

Oct 23 Topic: Lagrange’s Theorem

Oct 26 Topic: External Direct Products

Oct 28 Topic: Normal Subgroups

Oct 30 Topic: Factor Groups Assignment 8 DueNov 2 Topic: Factor Groups

Nov 4 Topic: The Fundamental Theorem of Finite Abelian Groups

Nov 6 Topic: Rings Assignment 9 DueNov 9 Topic: Rings

Nov 11 Topic: Integral Domains

Nov 13 Topic: Integral Domains Assignment 10 DueNov 16 Topic: Ideals

Nov 18 Topic: Factor Rings

Nov 20 Topic: Ring Homomorphisms Assignment 11 DueNov 23 - 27 Thanksgiving Break - NO CLASS

Nov 30 Topic: Polynomial Rings

Dec 2 Topic: Divisibility in Integral Domains

Dec 4 Final Exam Review Assignment 12 DueDec 7 Reading Day - NO CLASS

Dec 11 FINAL EXAM 8:00-10:00

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Why Study Modern Algebra?

The following is an excerpt from Joseph Gallian’s website on why modern algebra (also known as abstract algebra)is a valuable subject for math and math ed majors:

1. Even though many students take a course is discrete math where they study various proof techniques many ofthem seem not to absorb this material well. Abstract algebra provides them much more practice at this in adifferent context than discrete math does.

2. High school math teachers should be very adept at modular arithmetic. Cyclic groups is where they learn thiswell.

3. Group theory is the mathematics of symmetry–a fundamental notion in science, math and engineering. Forexample, the symmetry group of a molecule reveals some of its possible (or impossible) chemical properties.

4. There are many important practical applications of modular arithmetic that are best understood by viewingthe modular arithmetic in a group theory framework. Examples include the check digits on UPC codes onretail items, ISBN numbers on books, and credit card numbers. In many cases the check digit is the inverseof a weighted sum modulo an integer (10 in the case of a UPC number, 11 in the case of an ISBN number, 9in the case of Visa travelers checks).

5. Many games can be understood by viewing them as permutation groups. Two examples are the 15 puzzle andthe Rubik cube.

6. High school math teachers should be adept at looking at data and making plausible conjectures and general-izations. They should also teach their students to do this. This is a skill that can be learned with practice.Groups and rings provide abundant opportunities for developing this skill.

7. Many people are not comfortable with abstract concepts nor adept at abstract reasoning. The ability to thinkabstractly is a valuable asset. Abstract algebra helps develop this ability.

8. Abstract Algebra is an ideal capstone course for math ed majors and for those who will go on to grad schoolin math. Throughout the course they review things like 1-1 functions, onto functions (surprisingly few seniormath ed majors understand these ideas well); equivalence relations; basic concepts from linear algebra suchas how to multiply matrices, properties of determinants, how to compute a determinant, how to compute theinverse of a matrix, how to tell if a matrix has an inverse, linear transformations (which are group homomor-phisms); properties of complex numbers; properties of integers (Euclid’s lemma, division algorithm, criterionfor divisibility by 9 or 11 or 4); math induction (another important topic that many students do not seemto understand well when they begin an abstract algebra course–this is especially the case for statements thatdo not involve sums of series); and properties of polynomials (division algorithm, remainder theorem, factortheorem, number of zeros is at most the degree, unique factorization).

9. Doing well in an abstract algebra course is a confidence builder and sometimes causes students to think aboutgoing on the graduate school. I once had a student who did extremely well in abstract algebra who went tomedical school and now has a high position in the Center for Disease Control in Atlanta. About 20 years aftershe took the course I met her for dinner while I was at a meeting in Atlanta. I jokingly said to her ”Did youuse any abstract algebra in med school?” She immediately responded by saying ”Abstract algebra was veryvaluable to me in med school.” I asked how. She said that whenever she was taking a difficult course she saidto herself ”If I can get an A in abstract algebra I can get an A is any course.” She was perfectly serious. Manypeople who start out intending to be high school math teachers or even teach high school math for severalyears decide to go to grad school in math for an advanced degree (I and many others loved the course andwanted to go to grad school to continue studying the subject). Taking abstract algebra and doing well makessuch a move more likely and easier to do.

- Joseph Gallian

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